#8445743

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,5)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34)(35,41)(36,43)(37,40)(38,39)(42,45)(44,48)(46,47);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(19,29)(20,31)(24,28)(25,30)(34,39)(36,41)(40,47)(44,45);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36)(34,35)(37,45)(38,41)(39,43)(40,48)(42,47)(44,46);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37,38,46)(34,40,39,47)(35,42,43,48)(36,44,41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352939);